Problem type

Power of a complex number with De Moivre's theorem

What it solves: Raise a complex number to a whole-number power by going through polar form, where a power becomes one multiplication of the angle, and confirm the answer by multiplying the long way in exact arithmetic.
Subject: Complex Numbers
Topic: Length and angle
Also called: De Moivre, power of a complex number, raise to a power
Free example: Read the complete worked example

Femtobytes

(-sqrt(3) + i)^7 with De Moivre's theorem, step by step

femtobytes.com/solve/complex-power/-1_3_1_1_7

The problem

Work out this power and give the answer in the form $a + bi$ .

\left( -\sqrt{3} + i \right)^{7}

The result

\left( -\sqrt{3} + i \right)^{7} = 64 \sqrt{3} - 64 i

In polar form the answer has length $128$ and direction $-\frac{\pi}{6}$ .

checked 15 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.

Picture 1 of 2Step 1: See where the number lives

The number z drawn as an arrow from the origin. Its length is the modulus r and the angle it makes with the positive real axis is the argument, written with the Greek letter theta. Here

r = 2

and

\theta = \frac{5 \pi}{6}

See where the number lives

Why this step

A complex number is two real numbers travelling together, so it cannot sit on a number line; it needs a plane. Rectangular form $a + bi$ says "go $a$ across and $b$ up". Polar form describes the SAME point as "face this direction, walk this far". The reason to bother: when complex numbers are multiplied, their lengths multiply and their directions add. Rectangular form hides that completely; polar form makes it visible.

Here $z = -\sqrt{3} + i$ , so the real part is $a = -\sqrt{3}$ and the imaginary part is $b = 1$ . The symbol $i$ is the imaginary unit, the number with $i^2 = -1$ . On the plane, $z$ is the point $(a, b)$ : $a$ is read on the horizontal (real) axis and $b$ on the vertical (imaginary) axis.

The real part is negative and the imaginary part positive, so $z$ lies in the second quadrant: to the left and above. Its angle will be between $\frac{\pi}{2}$ and $\pi$ .

The number z drawn as an arrow from the origin. Its length is the modulus r and the angle it makes with the positive real axis is the argument, written with the Greek letter theta. Here

r = 2

and

\theta = \frac{5 \pi}{6}

Find the modulus: how far z is from the origin

Why this step

The arrow to $z$ is the long side of a right-angled triangle whose other sides are $\left| a \right|$ across and $\left| b \right|$ up. Pythagoras gives its length. We square $a$ and $b$ first, which also throws away their signs: a distance cannot care whether we went left or right.

\displaystyle r

=

\displaystyle \sqrt{\left( -\sqrt{3} \right)^{2} + 1^{2}}

The modulus

r = \left| z \right| = \sqrt{a^2 + b^2}

, by Pythagoras. checked

=

\displaystyle \sqrt{3 + 1}

Square each part. Squaring a square root undoes it, for example

\left( \sqrt{3} \right)^2 = 3

, and the square of a negative number is positive. checked

=

\displaystyle \sqrt{4}

Add. checked

=

\displaystyle 2

4

is a perfect square, so the root comes out exactly. checked

The same arrow, with the right-angled triangle it is the long side of. The side across is the real part and the side up is the imaginary part, so the modulus is the length Pythagoras gives from those two.

Power of a complex number with De Moivre's theorem

See where the number lives

Find the modulus: how far z is from the origin

Find the argument: which way z points

Write the polar form, and check it by turning it back

See what multiplying does in polar form

Apply De Moivre: raise the length, multiply the angle

Turn the answer back into the form a + bi

Check by multiplying the long way